Categorification and the quantum Grassmannian. (English) Zbl 1503.13012
In earlier work [Proc. Lond. Math. Soc. (3) 113, No. 2, 185–212 (2016; Zbl 1375.13033)], the same three authors explained how the cluster algebra structure on the homogeneous coordinate ring of a Grassmannian, as described by J. Scott [Proc. Lond. Math. Soc., III. Ser. 92, No. 2, 345–380 (2006; Zbl 1088.22009)], is additively categorified by the category of maximal Cohen-Macaulay modules over a certain Gorenstein order.
Among other properties reflecting the combinatorics of the Grassmannian cluster algebra, this category has ‘rank \(1\)’ indecomposable objects in bijection with Plücker coordinates, which are Ext-orthogonal if and only if the corresponding Plücker labels are non-crossing. This leads to a bijection between the cluster-tilting objects of the category (or at least those mutation equivalent to one for which all indecomposable summands have rank \(1\)) and the seeds in the Grassmannian cluster algebra. This bijection is compatible with mutation, and the quiver of a seed can be computed as the Gabriel quiver of the endomorphism algebra of the corresponding cluster-tilting object.
The Grassmannian coordinate ring admits a natural quantisation, and J. E. Grabowski and S. Launois [Proc. Lond. Math. Soc. (3) 109, No. 3, 697–732 (2014; Zbl 1315.13036)] show that the cluster algebra structure can be quantised compatibly. In the present paper, the authors show that additional quantum information – precisely, the powers of \(q\) appearing in quasi-commutation relations among quantum cluster variables – may be computed from the same category of Cohen-Macaulay modules which categorifies the unquantised cluster structure. Thus this same category, with no modification such as passing to graded modules, categorifies the quantum Grassmannian cluster algebra.
For rank \(1\) modules, the \(\kappa\)-invariant, which the authors introduce to compute the quasi-commutation power of the corresponding Plücker coordinates, may also be expressed as an invariant MaxDiag of a pair of Young diagrams, as in work of K. Rietsch and L. Williams [Duke Math. J. 168, No. 18, 3437–3527 (2019; Zbl 1439.14142)].
We note that while this paper yields a categorical interpretation of the quasi-commutation data (or \(L\)-matrix) for each seed in the quantum Grassmannian cluster algebra, it does not provide a quantum cluster character, computing expressions for quantum cluster variables as quantum Laurent polynomials in a chosen initial seed. This is well-known to be a difficult problem, especially for cluster algebras without acyclic seeds, as is the case for most Grassmannians, and for now remains open.
Among other properties reflecting the combinatorics of the Grassmannian cluster algebra, this category has ‘rank \(1\)’ indecomposable objects in bijection with Plücker coordinates, which are Ext-orthogonal if and only if the corresponding Plücker labels are non-crossing. This leads to a bijection between the cluster-tilting objects of the category (or at least those mutation equivalent to one for which all indecomposable summands have rank \(1\)) and the seeds in the Grassmannian cluster algebra. This bijection is compatible with mutation, and the quiver of a seed can be computed as the Gabriel quiver of the endomorphism algebra of the corresponding cluster-tilting object.
The Grassmannian coordinate ring admits a natural quantisation, and J. E. Grabowski and S. Launois [Proc. Lond. Math. Soc. (3) 109, No. 3, 697–732 (2014; Zbl 1315.13036)] show that the cluster algebra structure can be quantised compatibly. In the present paper, the authors show that additional quantum information – precisely, the powers of \(q\) appearing in quasi-commutation relations among quantum cluster variables – may be computed from the same category of Cohen-Macaulay modules which categorifies the unquantised cluster structure. Thus this same category, with no modification such as passing to graded modules, categorifies the quantum Grassmannian cluster algebra.
For rank \(1\) modules, the \(\kappa\)-invariant, which the authors introduce to compute the quasi-commutation power of the corresponding Plücker coordinates, may also be expressed as an invariant MaxDiag of a pair of Young diagrams, as in work of K. Rietsch and L. Williams [Duke Math. J. 168, No. 18, 3437–3527 (2019; Zbl 1439.14142)].
We note that while this paper yields a categorical interpretation of the quasi-commutation data (or \(L\)-matrix) for each seed in the quantum Grassmannian cluster algebra, it does not provide a quantum cluster character, computing expressions for quantum cluster variables as quantum Laurent polynomials in a chosen initial seed. This is well-known to be a difficult problem, especially for cluster algebras without acyclic seeds, as is the case for most Grassmannians, and for now remains open.
Reviewer: Matthew Pressland (Glasgow)
MSC:
| 13F60 | Cluster algebras |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 16G50 | Cohen-Macaulay modules in associative algebras |
| 20G42 | Quantum groups (quantized function algebras) and their representations |
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